Nuprl Lemma : r2-equidistant-implies
∀a,b:ℝ^2. (a ≠ b ⇒ (∀x:ℝ^2. (ax=bx ⇒ (∃t:ℝ. req-vec(2;x;vec-midpoint(a;b) + t*r2-perp(b - a))))))
Proof
Definitions occuring in Statement :
vec-midpoint: vec-midpoint(a;b),
r2-perp: r2-perp(x),
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec-mul: a*X,
real-vec-sub: X - Y,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-dist: d(x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
real-vec-sep: a ≠ b,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
vec-midpoint: vec-midpoint(a;b),
rneq: x ≠ y,
guard: {T},
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermSubtract: left "-" right,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
rtermMultiply: left "*" right,
pi2: snd(t),
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rv-congruent: ab=cd,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
real-vec-sub: X - Y,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B
Lemmas referenced :
rv-congruent_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
real-vec-sep_wf,
real-vec_wf,
int-to-real_wf,
real-vec-dist_wf,
rless_functionality,
req_weakening,
real-vec-dist-symmetry,
dot-product_wf,
real-vec-sub_wf,
vec-midpoint_wf,
rsub_wf,
req_functionality,
dot-product-linearity1-sub,
real-vec-mul_wf,
real-vec-add_wf,
rdiv_wf,
rless-int,
rless_wf,
rmul_wf,
dot-product-linearity2,
radd_wf,
rmul_functionality,
req_transitivity,
rsub_functionality,
dot-product-linearity1,
dot-product-comm,
radd_functionality,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermSubtract_wf,
rtermVar_wf,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
radd-preserves-req,
rminus_wf,
rinv_wf2,
itermMultiply_wf,
itermMinus_wf,
rinv-mul-as-rdiv,
rminus_functionality,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real-vec-dist-equal-iff,
rmul_preserves_req,
minus-one-mul-top,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rmul-rinv3,
int-rinv-cancel,
req-implies-req,
r2-dot-product-eq-0-iff-perp,
req-vec_wf,
r2-perp_wf,
req-vec_functionality,
req-vec_weakening,
real-vec-add_functionality,
req-vec_inversion,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
applyEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
closedConclusion,
inrFormation_alt,
independent_pairFormation,
imageMemberEquality,
baseClosed,
int_eqEquality,
minusEquality,
instantiate,
cumulativity,
intEquality,
equalityIstype,
sqequalBase
Latex:
\mforall{}a,b:\mBbbR{}\^{}2. (a \mneq{} b {}\mRightarrow{} (\mforall{}x:\mBbbR{}\^{}2. (ax=bx {}\mRightarrow{} (\mexists{}t:\mBbbR{}. req-vec(2;x;vec-midpoint(a;b) + t*r2-perp(b - a))))))
Date html generated:
2019_10_30-AM-11_32_32
Last ObjectModification:
2019_04_02-PM-04_21_24
Theory : reals!model!euclidean!geometry
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